Chapter 7: Ecosystems

7.3 Material Cycling

7.3.1 Linear trophic interactions

Nutrient-plant system: losses from the linear food-chain are collected in a nutrient compartiment from which material for primary production must be drawn. Nutrient-plant example: Limiting nutrient \(N(t)\) and plant functional group \(P(t)\) characterized by the limiting nutrient density (instead of carbon). System is closed, so any nutrient taken up by the plants (\(\alpha_p N P\)) is lost to the free nutrient pool, and returned via death and excretion, immediately added back to the pool (\(\delta_p P\)).

\[\frac{dP}{dt} = \alpha_p N P - \delta_p P, \quad \frac{dN}{dt} = \delta_p P - \alpha_p P N\] Note that \[\frac{dP}{dt} + \frac{dN}{dt} \frac{d}{dt}(P + N) = 0\] so we can define \(S \equiv P + N\) as the total nutrient in the (closed) system.

Writing \(N\) as \(S - P\) and subbing into the \(dP/dt\) eqn: \[\frac{dP}{dt} = (\alpha_p S - \delta_p)P\left[1 - \frac{\alpha_p P}{\alpha_p S - \delta_p} \right]\] then consider that if we define \(r \equiv \alpha_p S - \delta P\) and \(K \equiv r_p / \alpha_p = S - \delta_p / \alpha_p\) then we have a standard logistic regression model.

Nutrient-plant-herbivore-consumer system

Now we add an herbivore and consumer level.

  • \(N = S - P - C - H\)
  • \(\frac{dP}{dt} = \alpha_p N P - \delta_p P - \alpha_h P H\)
  • \(\frac{dH}{dt} = \alpha_h P H - \delta_h H - \alpha_c H C\)
  • \(\frac{dC}{dt} = \alpha_c H C - \delta_c C\)

see table 7.2 for steady states of this system.

NPHC dynamics

7.3.2 Type II trophic interactions

Nutrient-plant system: losses from the linear food-chain are collected in a nutrient compartiment from which material for primary production must be drawn. Nutrient-plant example: Limiting nutrient \(N(t)\) and plant functional group \(P(t)\) characterized by the limiting nutrient density (instead of carbon). System is closed, so any nutrient taken up by the plants (\(\alpha_p N P\)) is lost to the free nutrient pool, and returned via death and excretion, immediately added back to the pool (\(\delta_p P\)).

Nutrient-plant-herbivore system

Now we add an herbivore level.

  • \(N = S - P - C - H\)
  • \(\frac{dP}{dt} = \alpha_p N P - \delta_p P - \frac{\alpha_h P H}{P + P_0}\)
  • \(\frac{dH}{dt} = \frac{\alpha_h P H}{P + P_0} - \delta_h H\)

Stable steady states:

  • \(P^* = \frac{\delta_h P_0}{\alpha_h - \delta_h}\)
  • \(H^* = \frac{\alpha_p(S - P^*) - \delta_p}{\alpha_p + \alpha_h / (P^* + P_0)}\)

NPH II dynamics